The second wave of unaccompanied illegal immigrant children has begun, with more than 3,000 of them surging across the Mexican border into the U.S. last month — the highest rate since the peak of last summer’s crisis and a warning that another rough season could be ahead.

Immigration officials warned that they expected another surge as the weather improved. Although the numbers are down some 40 percent compared with last year’s frenetic pace that sparked a political crisis for the Obama administration, fiscal year 2015 is shaping up to mark the second-biggest surge on record.

Authorities report having captured 15,647 children traveling without parents who tried to jump the border in the first six months of the fiscal year. Through this point in 2014, they had apprehended 28,579.

Just as worrisome is the rate of whole families — usually mothers with young children — who are crossing. So far this fiscal year, authorities have captured 13,911 “family units,” down 30 percent from last year.

“These statistics show that the surge of illegal arrivals from Central America was never really over,” said Jessica Vaughan, policy studies director at the Center for Immigration Studies.

Among those is the policy that requires children from Central America to be released into the U.S. rather than quickly returned to their home countries. Once released, those children usually fail to appear for deportation proceedings.

The Congressional Research Service told Congress in late March that 62 percent of the children failed to show up for their cases before immigration judges from July through February. All of them were ordered deported, but the workload of officials made deportation unlikely in most cases.

The Obama administration last year initially blamed bad economies and growing gang violence in Honduras, El Salvador and Guatemala for sparking the surge, but later acknowledged that human traffickers were marketing the journey by pointing out a loophole in U.S. immigration system that requires non-Mexican children to be released into the U.S. while they await final immigration decisions. That gives them a chance to abscond and disappear into the shadows with the more than 11 million other illegal immigrants in the country.

“Those are pretty bad outcomes for immigration hearings,” Ms. Vaughan said. “Lots of no-shows and few people getting relief. These statistics show that the administration’s response has been a failure.”

U.S. Customs and Border Protection, the agency that oversees the Border Patrol, didn’t return a message for comment Monday, but Homeland Security spokeswoman Marsha Catron said the Obama administration has poured manpower and resources into trying to secure the border and vowed to update plans “to ensure we are as prepared as possible for any potential scenario.”

She said agencies now have more detention beds to hold families, which means they can be deported at a faster pace. She also pointed to a new program that will allow Central American children to apply from their home countries to join their parents in the U.S., including those who are gaining tentative legal status under Mr. Obama’s deportation amnesty.

Analysts have been debating what other steps are needed to stem another surge this year.

Adolfo F. Franco, a former official at the U.S. Agency for International Development overseeing Latin America and the Caribbean, urged Congress to try to reverse Mr. Obama’s deportation amnesty. He said the amnesty serves as further enticement for those who believe they can stay if they get into the U.S.

“There is a sense that the law in the U.S. has changed, and therefore it is easier to come to the United States and ultimately get a work permit and Social Security number,” Mr. Franco told the Senate Homeland Security and Governmental Affairs Committee on March 25. “That is the driver and that is the pull factor, not the other isolated cases.”

Eric L. Olson, who studies Latin America for the Woodrow Wilson International Center for Scholars, insisted that violence in Central American countries is the major factor pushing people out of their homes and sending them north. He said his travels to the region suggest that many families face difficult choices: try to work out an agreement with gangs to leave them alone, watch their children either join or be targeted by the gangs, or flee.

“It’s not surprising that many decide to flee,” he said.

If a lax U.S. enforcement policy were the pull factor, Mr. Olson said, it also would also be enticing Mexicans and others from Central America, but the surge has been chiefly from El Salvador, Guatemala and Honduras.

Of the 15,647 unaccompanied minors caught since Oct. 1 by U.S. authorities, 2,788 were from El Salvador, 5,465 were Guatemalan, 1,549 were Honduran and 5,572 were Mexican. The number from Mexico is in line with previous years’ rates.

Mr. Franco said the U.S. must change its law to speed deportations of Central Americans the same way it allows for quick turnaround of Mexicans caught trying to cross the border. Almost all Mexican juveniles are quickly sent home, which deters a much larger wave, analysts said.

Obama administration officials initially expressed support for those kinds of changes but backed off last year. Instead, they asked for more money to boost the economies of Central America and tried to exert diplomatic pressure on governments to try to keep their citizens at home. The U.S. government also paid for public service announcements discouraging the dangerous journey.

The administration has claimed credit for lower numbers this year, but it is still struggling to deal with those who do arrive.

Under federal law, children in the U.S. illegally who aren’t immediately returned home are to be transferred to social workers at the Health and Human Services Department, which holds them until they can be placed with relatives or foster family sponsors.

Sen. Ted Cruz, Texas Republican, last week asked HHS to divulge details of sexual abuse or assaults on minors housed at facilities run or contracted by HHS.

He also said the government isn’t adequately weeding out adults posing as minors or those with criminal records or gang ties, and isn’t fully vetting the relatives or other sponsors who house the children.

“I have also been told that, pursuant to department policy and with a few minor exceptions, the department conducts little or no post-release monitoring of [unaccompanied alien children], which raises many questions about the safety and security of minors that have been and will be released from your care,” Mr. Cruz wrote in a letter demanding information.